you are standing on one side of the river. Figure out the width of the river. All calculations have to be made on the side you are standing on. You cant step into the river or cross it. You may use the 2 laws ( sine and cos ). no guessing allowed. has to be accurate.
You know how high you are. You know or can measure the angle downward from the horizontal to the opposite side (use some sort of survey telescope, but you have to determine that angle.)
You then can easily determine it. You know one side, and an angle.
Tangent would be nice, and tangent is just your height/width. and you solve for width.
There is another way, more difficult.
i don't understand what you mean , simpler terms?
draw the triangle from a point on the opposite shore to your feet, then one leg up, your height. You measure the depression angle from the horizon to the point on the other shore. So in the triangle, that is the same angle at the base of the triangle on the other side.
Tan angle= your height/width
To figure out the width of the river without stepping into or crossing it, you can use the concept of triangulation and the laws of trigonometry, specifically the sine and cosine functions.
Here's a step-by-step explanation of how to do it:
1. Find a fixed point on your side of the river that provides a clear line of sight to the opposite side. This point should be far enough from the river to have a wide-angle view of the opposite bank and ideally, have an elevated position.
2. Measure the distance from your position to the fixed point and mark it as "a."
3. Take note of a prominent object or point on the opposite bank of the river that aligns with your line of sight. This will act as a reference point and will later help you calculate the width of the river.
4. Walk a known distance perpendicular to your line of sight, away from the river.
5. Measure the horizontal distance you have walked perpendicularly from the fixed point and mark it as "b."
6. Next, measure the angle between your original line of sight and the new line connecting the fixed point to your new position. This angle is formed by the perpendicular line you walked and the line connecting the fixed point to the reference point on the opposite bank.
7. With the measured angle, apply the cosine function to calculate the adjacent side, which represents the width of the river. The formula would be:
Cos(angle) = b / adjacent side
Rearranging this equation, we get:
adjacent side = b / cos(angle)
8. Plug in the measured "b" value and the calculated cosine of the angle into the formula to find the width of the river.
This method allows you to calculate the width of the river accurately without physically crossing it or stepping into the water, using only trigonometry and measurements made on your side of the river.